Nuprl Lemma : total-function-rational-approx
∀f:ℝ ⟶ ℝ. ∀y:ℝ. ((∀x,y:ℝ. ((x = y) ⇒ (f[x] = f[y]))) ⇒ lim n→∞.f[(y within 1/n + 1)] = f[y])
Proof
Definitions occuring in Statement :
converges-to: lim n→∞.x[n] = y,
rational-approx: (x within 1/n),
req: x = y,
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
real: ℝ,
nat_plus: ℕ+,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
less_than': less_than'(a;b),
true: True,
so_apply: x[s]
Lemmas referenced :
total-function-limit,
rational-approx_wf,
decidable__lt,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
less_than_wf,
nat_wf,
rational-approx-converges-to,
all_wf,
real_wf,
req_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
isectElimination,
setElimination,
rename,
because_Cache,
dependent_set_memberEquality,
addEquality,
natural_numberEquality,
productElimination,
unionElimination,
independent_pairFormation,
voidElimination,
independent_functionElimination,
independent_isectElimination,
applyEquality,
isect_memberEquality,
voidEquality,
intEquality,
minusEquality,
functionEquality,
functionExtensionality
Latex:
\mforall{}f:\mBbbR{} {}\mrightarrow{} \mBbbR{}. \mforall{}y:\mBbbR{}. ((\mforall{}x,y:\mBbbR{}. ((x = y) {}\mRightarrow{} (f[x] = f[y]))) {}\mRightarrow{} lim n\mrightarrow{}\minfty{}.f[(y within 1/n + 1)] = f[y])
Date html generated:
2017_10_03-AM-10_21_53
Last ObjectModification:
2017_06_30-PM-04_15_10
Theory : reals
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